I'm running some JJ E34Ls in mine right now. I think it tightens up the bass and I like the way the drive sounds. The 6L6s seem to go deeper but the 34s seem to make you feel the low end more. It is like comparing two really nice wines. They are both good, just different.
I do not notice that much difference in the clean channel, maybe the 6L6s are a little more honey poured over them and the EL34s have just a slight more edge, but in the LSC they both sound great.
In the crive channel, I like how the EL34s hold on to the mids and low end and give you a great crunch. It is the tonality of a Marshall, but with the low end holding together, not all loose like on a Marshall. It is the Mesa version of a Brit EL34.
I like the effect of the Thick switch with the EL34s more than with the 6L6s. I think that particular shift in mids really compliments those tubes. When I play the power chords of Sunshine of Your Love or Cocain, you get some real girth. Back off the tone a little and Whoa!! I am thinking this setup deserves a 2x12 extension cab!
I would give them a whirl. I don't think you'll throw the 6L6s away, but you'll have an option on tone that few amps can match.